Circle P has diameter CD. Point B is on the circle such that mBPC = 30 degrees. Point A
is on the circle such that AD is parallel to PB. What is the degree measure of arc ABC?
The answer is 60 degrees
Since AD is parallel to PB, we can apply the alternate angle theorem to determine the degree measure of arc ABC. According to the alternate angle theorem, if two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
In this case, angle BPC is a central angle, and angle BAC is an inscribed angle that intercepts the same arc. Therefore, the measure of arc ABC is equal to the measure of angle BPC, which is 30 degrees.
To find the degree measure of arc ABC, we need to use the properties of angles and arcs in a circle.
Given:
- Circle P with diameter CD
- ∠BPC = 30 degrees
- AD is parallel to PB
Since CD is a diameter of the circle, we know that angle ∠BPC is inscribed in the circle and its intercepted arc is CD.
From the inscribed angle theorem, we know that the measure of an inscribed angle is equal to half the measure of its intercepted arc. Therefore, the intercepted arc CD measures 2*∠BPC = 2*30 = 60 degrees.
Now, since AD is parallel to PB, the angle between them (angle BAD) is equal to the alternate angle ∠BPC, which is 30 degrees.
Arc ABC is intercepted by angle BAD. According to the inscribed angle theorem, the measure of arc ABC is equal to the measure of angle BAD. Therefore, arc ABC measures 30 degrees.
In conclusion, the degree measure of arc ABC is 30 degrees.